The video display you describe suggests the notorious eMac capacitor problem. There's oodles of info about it on this and other Mac forums/sites.
Turn your eMac off completely, but do not disconnect the power cord. Tip the eMac on its screen face on a towel (or similar). Open the hatch on the underside, that'll give you access to the RAM slots. Look beyond those to the capacitors on the logic board that are visible (about 5 of them).
Do you see any bulging of the capacitor caps, or brown/yellow gunge seeping therefrom? If so, you have a major hardware issue to resolve. Post back once you've carried out the inspection, and we can then advise you further.